A note on a finite element method dealing with corner singularities

Recently the first author and his coworker report a new finite element method for the Poisson equations with homogeneous Dirichlet boundary conditions on a polygonal domain with one re-entrant angle [7]. They use the well-known fact that the solution of such problem has a singular representation, deduce a well-posed new variational problem for regular part of solution and an extraction formula for the so-called stress intensity factor using two cut-off functions.They use Fredholm alternative and Gårding’s inequality to establish the well-posedness of the variational problem and finite element approximation, so there is a maximum bound for meshh theoretically, although the numerical experiments shows the convergence for every reasonable size ofh. In this paper we show that the method converges for everyh with reasonable size by imposing a restriction to the support of the extra cut-off function without using Gårding’s inequality. We also give error analysis with similar results.